MCPCalc

Dice Probability Calculator

Calculate exact probabilities for N dice with S sides for exact/at-least/at-most/range outcomes.

Scratchpad (not saved)

What This Calculator Does

This calculator builds the full discrete distribution for the sum of fair dice, then reports exact, cumulative, and range probabilities from that distribution. It works for tabletop games, probability homework, and any scenario where the sum of multiple independent dice matters.

It combines Number of Dice, Sides Per Die, Mode, Target Sum to estimate Selected Probability, Selected Probability (Decimal), Exact Probability.

Formula & Method

Core equations: Total outcomes for NN fair SS-sided dice is SNS^N. Exact sum probability is P(X=t)=#{outcomes summing to t}SNP(X=t)=\frac{\#\{\text{outcomes summing to }t\}}{S^N}, where counts are obtained by dynamic-programming convolution of die PMFs. Then P(X≥t)=∑x=tNSP(X=x)P(X\ge t)=\sum_{x=t}^{NS}P(X=x), P(X≤t)=∑x=NtP(X=x)P(X\le t)=\sum_{x=N}^{t}P(X=x), and range probability P(a≤X≤b)=∑x=abP(X=x)P(a\le X\le b)=\sum_{x=a}^{b}P(X=x). Distribution moments shown are μ=N(S+1)2\mu=\frac{N(S+1)}{2} and σ=N(S2−1)12\sigma=\sqrt{\frac{N(S^2-1)}{12}}.

Notation used in the formulas: RR = Selected Probability; x1x_{1} = Number of Dice; x2x_{2} = Sides Per Die; x3x_{3} = Mode; x4x_{4} = Target Sum; x5x_{5} = Range Max (for Range mode).

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it when you need exact dice odds instead of approximations, especially for custom dice pools, threshold checks, or balancing tabletop mechanics around a target success band.

Reference Book

Introduction to Probability

Second Edition

Joseph K. Blitzstein and Jessica Hwang · Chapman and Hall/CRC

A widely used probability text for counting, conditioning, expected value, core distributions, and simulation.

View Book

Inputs Used

  • Number of Dice: Used directly in the calculation.
  • Sides Per Die: Used directly in the calculation.
  • Mode: Used directly in the calculation.
  • Target Sum: Used directly in the calculation.
  • Range Max (for Range mode): Used directly in the calculation.

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